Questions tagged [combinatorics]

For questions about the study of finite or countable discrete structures, especially how to count or enumerate elements in a set (perhaps of all possibilities) or any subset. It includes questions on permutations, combinations, bijective proofs, and generating functions.

The goal of the four fours puzzle is to represent each natural number using four copies of the digit $4$ and common mathematical symbols.
For example, $165=\left(\sqrt{4} + \sqrt{\sqrt{{\sqrt{4^{4!}}}...

This page gives a few examples of Venn diagrams for 4 sets. Some examples:
Thinking about it for a little, it is impossible to partition the plane into the $16$ segments required for a complete $4$-...

This is a neat little problem that I was discussing today with my lab group out at lunch. Not particularly difficult but interesting implications nonetheless
Imagine there are a 100 people in line to ...

I have $n$ people seated around a circular table, initially in arbitrary order. At each step, I choose two people and switch their seats. What is the minimum number of steps required such that every ...

Popular mathematics folklore provides some simple tools
enabling us compactly to describe some truly enormous
numbers. For example, the number $10^{100}$ is commonly
known as a googol,
and a googol
...

What is your recommendation for an in-depth introductory combinatoric book? A book that doesn't just tell you about the multiplication principle, but rather shows the whole logic behind the questions ...

You are a student, assigned to work in the cafeteria today, and it is your duty to divide the available food between all students. The food today is a sausage of 1m length, and you need to cut it into ...

This is a soft question, but I've tried to be specific about my concerns. When studying basic combinatorics, I was struck by the fact that it seems hard to verify if one has counted correctly.
It's ...

How many ways can I write a positive integer $n$ as a sum of $k$ nonnegative integers up to commutativity?
For example, I can write $4$ as $0+0+4$, $0+1+3$, $0+2+2$, and $1+1+2$.
I know how to find ...

This combinatorial question has a musical motivation, which I provide below using as little musical jargon as I can. But first, I'll present a purely mathematical formulation for those not interested ...

Suppose a biased coin (probability of head being $p$) was flipped $n$ times. I would like to find the probability that the length of the longest run of heads, say $\ell_n$, exceeds a given number $m$, ...

A student has to pass a exam, with $k2^{k-1}$ questions to be answered by yes or no, on a subject he knows nothing about. The student is allowed to pass mock exams who have the same questions as the ...

$7$ fishermen caught exactly $100$ fish and no two had caught the same number of fish. Prove that there are three who have captured together at least $50$ fish.
Try: Suppose $k$th fisher caught $r_k$ ...

I had seen this problem a long time back and wasn't able to solve it. For some reason I was reminded of it and thought it might be interesting to the visitors here.
Apparently, this problem is from a ...

Suppose we have a power tower consisting of $2$ occurring $n$ times:
$$\huge2^{2^{2^{.^{.^{.^{2}}}}}}$$
How many values can we generate by placing any number of parenthesis?
It is fairly simple for ...

Divide a square of side $n$ into any number of non-congruent rectangles. If all the sides are integers, what is the smallest possible difference in area between the largest and smallest rectangles?
...

I'm not a very smart man. I'm trying to count how many years I've been working at my new job. I started in May 2011.
If I count the years separately, I get that I've worked 4 years - 2011 (year 1), ...

The Catalan numbers have a reputation for turning up everywhere, but the occurrence described below, in the analysis of an (incorrect) algorithm, is still mysterious to me, and I'm curious to find an ...

A common way to define a group is as the group of structure-preserving transformations on some structured set. For example, the symmetric group on a set $X$ preserves no structure: or, in other words,...

I have the following (real world!) problem which is most easily described using an example.
I ask my six students when the best time to hold office hours would be. I give them four options, and say ...

I saw this question today, it asks how many triangles are in this picture.
I don't know how to solve this (without counting directly), though I guess it has something to do with some recurrence.
How ...